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Fractional calculus of Weyl algebra and Fuchsian differential equations. (English) Zbl 1269.34003

MSJ Memoirs 28. Tokyo: Mathematical Society of Japan (ISBN 978-4-86497-016-7/pbk). xix, 203 p. (2012).
The book under review makes a unifying approach to the study of algebraic and analytic structures of a general Fuchsian differential equation on the Riemann sphere, namely, a linear differential equation with rational coefficients whose singularities are regular. The key to developing the theory is the fractional calculus on the Weyl algebra.
Consider a system of Fuchsian equations of the Schlesinger canonical form \[ \frac{du}{dx} = \sum_{j=1}^p \frac{A_j}{x-c_j} u, \tag{1} \] where \(A_1, \dotsc, A_p\) are constant square matrices. The local system associated with (1) or system (1) itself is said to be rigid if it is uniquely determined by the local monodromies of (1) around the singularities \(c_j\). N. M. Katz [Rigid local systems. Princeton, NJ: Princeton Univ. Press (1996; Zbl 0864.14013)] and M. Dettweiler and S. Reiter [J. Algebra 318, No. 1, 1–24 (2007; Zbl 1183.34137); J. Symb. Comput. 30, No. 6, 761–798 (2000; Zbl 1049.12005)] proved that every irreducible rigid system is changed into the trivial equation \(du/dx=0\) by successive application of operations called an addition and a middle convolution, respectively. Here, the addition is the gauge transformation \(u(x)\mapsto (x-c)^{\lambda}u(x)\), and the middle convolution is essentially the Euler transformation \[ u(x) \mapsto \frac 1{\Gamma(\mu)} \int^x_c u(t) (x-t)^{\mu-1}\, dt. \] V. P. Kostov posed the problem called the Deligne-Simpson problem: find a condition on the matrices \(B_0, B_1, \dotsc, B_p\) of Jordan canonical form that guarantees the existence of an irreducible system of the form (1) with \(A_j\) conjugate to \(B_j\) for \(j=0,1, \dotsc, p\), where \(A_0:= -(A_1+\dotsb +A_p)\); and gave an answer under a certain genericity condition [in: S. Dimiev (ed.) et al., Perspectives of complex analysis, differential geometry and mathematical physics. Proceedings of the 5th international workshop on complex structures and vector fields, St. Konstantin, Bulgaria, September 3–9, 2000. Singapore: World Scientific. 1–35 (2001; Zbl 0999.20039); J. Algebra 281, No. 1, 83–108 (2004; Zbl 1066.15016)].
In this book, the author develops an extensive study on a general Fuchsian equation which is not necessarily rigid; the study includes generalizations of the results above established by examining the fundamental spectral type and by constructing a universal Fuchsian differential operator that may have accessory parameters. In particular, it is shown that there exists a single ordinary differential equation without apparent singularities realizing a given rigid local system; which is an affirmative answer to the question of Katz. Furthermore, for a general Fuchsian equation, the author discusses reducibility, integral expressions of solutions, contiguity relations, and connection coefficients.
This book consists of 14 chapters.
The first chapter gives the framework in which the general Fuchsian equation is handled. Let \(W[x]\) be the Weyl algebra, namely, the ring of differential operators with polynomial coefficients. Since any linear ordinary differential equation is isomorphic to a single differential equation, in considering a Fuchsian equation, we may restrict the study to the category of single equations of the form \(Pu=0\) with \(P\in W[x]\). The fractional operations are introduced as operations on the elements of \(W[x]\), which also work on solutions of \(Pu=0\) as certain transformations. The gauge transformation (or the addition) and the Euler transformation (or the middle convolution) are composites of some of them.
In Chapter 2, the confluence of a Fuchsian equation is illustrated by using the Weyl algebra \(W[x,\xi]\) with some parameters \(\xi=(\xi_j)\). Related to the confluence procedure, versal operators are given for the Gauss hypergeometric equation, the Jordan-Pochhammer equation, the Kummer equation and the Hermite-Weber equation. After this chapter, every equation of the form \(Pu=0\) is of Fuchsian type.
Chapter 3 is devoted to examining the actions of fractional operations on series expansions and contiguity relations for solutions of \(Pu=0\).
In Chapter 4, at each regular singularity of \(Pu=0\), the set of generalized characteristic exponents \(\{ [\lambda_1]_{(m_1)}, \dotsc, [\lambda_N]_{(m_N)} \}\) with \([\lambda]_{(k)} ={}^T(\lambda, \lambda+1, \dotsc, \lambda+k-1)\) is defined, which appears in the generalized Riemann scheme for the equation \[ \left\{ \begin{matrix} \begin{matrix} x=c_0=\infty & c_1 & \dots & c_p \\ [\lambda_{0,1}]_{(m_{0,1})} & [\lambda_{1,1}]_{(m_{1,1})} & \dots & [\lambda_{p,1}]_{(m_{p,1})} \\ \vdots & \vdots & & \vdots \\ [\lambda_{0,n_0}]_{(m_{0,n_0})} & [\lambda_{1,n_1}]_{(m_{1,n_1})} & \dots & [\lambda_{p,n_p}]_{(m_{p,n_p})} \end{matrix} & ;\,\, x \end{matrix} \right\} \] satisfying the Fuchs relation \[ \sum^p_{j=0}\sum^{n_j}_{\nu=1}m_{j,\nu}\lambda_{j,\nu}- n + \frac {\operatorname{idx}\mathbf{m}}{2}=0, \quad \operatorname{idx}\mathbf{m} := \sum^p_{j=0}\sum^{n_j}_{\nu=1}m_{j,\nu}^2 -(p-1)n^2. \] Here, \(n=m_{j,1}+ \dotsb +m_{j,n_j}\) is the order of \(P\), and \(\operatorname{idx}\mathbf{m} \) coincides with the index of rigidity defined by Katz. The \((p+1)\)-tuple \(\mathbf{m}:=((m_{j,\nu})_{\nu=1,\dotsc, n_j}) _{j=0,\dotsc,p}\) is called the spectral type of \(P\) (or of the Riemann scheme for \(P\)). The tuple \(\mathbf{m}\) is said to be realizable (respectively, irreducibly realizable) if, for generic \(\lambda_{j,\nu}\) satisfying the Fuchs relation, there exists a (respectively, irreducible) Fuchsian equation \(Pu=0\) with the Riemann scheme above.
The addition and the middle convolution work within the category of Fuchsian differential operators.
Chapter 5 is concerned with the actions of these operations changing the Riemann scheme for \(Pu=0\). Consequently, we may find the reduction procedure for the spectral type of \(P\) keeping the index of rigidity invariant.
The Deligne-Simpson problem is solved in Chapter 6, which gives a condition for a given tuple \(\mathbf{m}\) to be realizable (respectively, irreducibly realizable). Moreover, when \(\mathbf{m}\) is realizable, we may explicitly construct a differential operator \(P_{\mathbf{m}}\) having the Riemann scheme with the spectral type \(\mathbf{m}\), which is called a universal Fuchsian differential operator. In general \(P_{\mathbf{m}}\) is parametrized by accessory parameters \(g_1, \dotsc, g_N\), where \(N=1-(1/2)\operatorname{idx} \mathbf{m}\) if \(\mathbf{m}\) is irreducibly realizable. The tuple \(\mathbf{m}\) is said to be rigid if \(\mathbf{m}\) is irreducibly realizable and if \(N=0\); which means that \(P_{\mathbf{m}}\) is free from accessory parameters if \(\mathbf{m}\) is rigid. The result in this chapter also gives an affirmative answer to the question of Katz mentioned earlier.
Chapter 7 describes the combinatorial aspect of transformations of the spectral type \(\mathbf{m}\) of \(P\) induced from fractional operations by using the terminology of a Kac-Moody root system. The fractional operation corresponds to the action of a reflection on a root lattice. The universal equation \(P_{\mathbf{m}}=0\) or \(\mathbf{m}\) is said to be fundamental if it cannot be reduced to a lower order equation by any fractional operation. It is shown that there exist only a finite number of fundamental tuples such that \(\operatorname{idx}\mathbf{m}\) equals a fixed number; which generalizes the result of Katz concerning an irreducible rigid equation.
In Chapter 8, for a local solution of \(P_{\mathbf{m}}u=0\) around a singularity with simple characteristic exponents, an integral expression and a power series expansion are derived from the results in Chapters 3, 5 and 6. When \(\mathbf{m}\) is rigid, the integrand is a finite product of powers of linear polynomials.
Chapter 9 reviews some known results on the monodromy of a Fuchsian equation from the view point of fractional operations.
The reducibility of the universal Fuchsian differential operator \(P_{\mathbf{m}}\) is discussed in Chapter 10, which gives a necessary and sufficient condition on characteristic exponents so that the monodromy of \(P_{\mathbf{m}}u=0\) with the rigid spectral type \(\mathbf{m}\) is irreducible.
In Chapter 11, by constructing shift operators between rigid Fuchsian equations, a contiguity relation for certain solutions is obtained, which is a generalization of \[ c(F(a,b+1,c;x)-F(a,b,c;x)) = ax F(a+1,b+1,c+1;x) \] for the Gauss hypergeometric function. Concerning shift operators some more results are given.
Chapter 12 deals with the connection problem for solutions of \(P_{\mathbf{m}}u=0\). The change of the connection coefficient under a certain fractional operation is described without supposing the rigidity. For a rigid equation with simple characteristic exponents, the connection coefficient is expressed in terms of a quotient of gamma functions. When \(p=2\), this gives a generalization of the Gauss summation formula \[ F(a,b,c; 1)= \frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}. \] Another procedure to calculate the connection coefficient is also proposed, which is based on the calculation of its zeros and poles.
Chapter 13 gives many examples explaining the fractional calculus and giving concrete results. Among others, all the fundamental tuples \(\mathbf{m}\) satisfying \(\operatorname{idx}\mathbf{m} =0\), \(-2\), \(-4\), \(-6\) are listed.
In the final chapter, several related problems are posed.
In this book, intrinsic arguments are avoided if possible. Most results are concretely described and can be implemented in computer programs. This book is recommended to both specialists and graduate students who are interested in the related fields.

MSC:

34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations
34M35 Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms
34A05 Explicit solutions, first integrals of ordinary differential equations
34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
34M40 Stokes phenomena and connection problems (linear and nonlinear) for ordinary differential equations in the complex domain
34M50 Inverse problems (Riemann-Hilbert, inverse differential Galois, etc.) for ordinary differential equations in the complex domain

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